Descriptive statistics: central tendency, variability, the normal curve
Last reviewed September 2026 by Dr. Andrew Wichterman · Aligned to the current NCE blueprint. Valid for exams through June 30, 2027.
AI-narrated overview made with Google NotebookLM. The written lesson below is by Dr. Andrew Wichterman.
You have 30 scores in front of you. Where is the middle, and how spread out are they? That is all descriptive statistics is. This lesson explains mean, median, mode, skew, standard deviation and correlation from scratch, with pictures, and then shows how the exam asks about each.
Try to answer each from memory first, even if you have to guess. Guessing before reading makes the answer stick; reveal only after you have tried.
Start here: describing a pile of numbers
Imagine you gave a 25-question anxiety quiz to 30 clients and now have 30 scores in front of you. Nobody wants to read 30 numbers. You want to say two things about the pile: where is the middle, and how spread out are the scores? "Descriptive statistics" is just the name for the handful of numbers that answer those two questions. That is the whole lesson. Everything else here is vocabulary for those two ideas and a picture of what the pile usually looks like.
Where is the middle? Three answers
There are three different ways to say "the typical score," and they can give different answers.
Mean is the ordinary average: add up all the scores and divide by how many there are. It uses every score, which is its strength and its weakness, because one very unusual score can drag it around.
Median is the middle score when you line them all up from lowest to highest. Half the scores are above it and half below. If there is an even number of scores, it is halfway between the two middle ones. The median does not care how extreme the highest or lowest scores are, only that they are on one side or the other.
Mode is the most common score, the one that shows up most often. A set of scores can have two modes (bimodal) or none. It is the only "middle" you can use with nominal data, because you cannot average names but you can count which one appears most.
When the pile leans: skew
If you draw the pile of scores as a curve, a perfectly balanced pile is the bell-shaped normal curve, and the mean, median and mode all sit in the same spot in the middle. When the pile is lopsided, it is called skewed, and the three measures pull apart. The rule that trips people up: skew is named for the direction of the tail, not the hump.
Positive skew: the tail points right, toward high scores. Most people scored low and a few scored very high. Income is the classic example: most people earn moderate amounts and a few earn millions. Mean is greater than the median. Negative skew: the tail points left. Most people scored high and a few scored very low, like an easy test. Mean is less than the median. In both cases the mode is at the hump and the mean is out toward the tail, with the median in between.
How spread out? Three answers
Two groups can have exactly the same average and be completely different. Five students who all scored about 70 and five students who scored 50, 60, 70, 80 and 90 both average 70. The second group is far more spread out. "Variability" is the family of numbers that measure that spread.
Range is the simplest: highest score minus lowest score. Class A has a range of 4 (72 − 68); Class B has a range of 40. Quick, but fragile, because it depends entirely on the two most extreme scores.
Variance measures how far, on average, scores are from the mean, but it does so in squared units, which makes it hard to interpret directly. You mostly need to know that variance is the standard deviation squared.
Standard deviation (SD) is the square root of the variance, which puts it back into the original units. Think of it as "the typical distance of a score from the mean." If the SD of an IQ test is 15, a typical person is about 15 points from the average of 100. The SD is the most important number in this lesson because the normal curve and every standard score are built on it.
| Measure | Plain meaning | When you would use it |
|---|---|---|
| Range | Highest minus lowest | A quick, rough sense of spread |
| Interquartile range | The range of the middle 50% of scores (75th percentile minus 25th) | Spread that ignores extreme scores; pairs with the median |
| Variance | Average squared distance from the mean | Mostly a stepping stone to the SD; used inside other statistics |
| Standard deviation | Typical distance from the mean, in the original units | The standard measure of spread; the basis of z-scores and the normal curve |
The normal curve, again
Lesson 5 introduced the bell-shaped normal curve and the 68-95-99.7 rule. It belongs here too, because it is a description of how scores usually pile up: mean, median and mode in the center, about 68 percent of people within one standard deviation of the mean, about 95 percent within two, about 99.7 percent within three. If you have not read lesson 5 yet, the picture and the conversion row are there.
For this lesson, the important connection is that the SD is the unit the normal curve is measured in. "One standard deviation above the mean" is the same position on every normal curve, whether the test uses IQ points, T-scores or anything else. That is what lets you convert between score types.
Do two things move together? Correlation
Descriptive statistics can also describe the relationship between two sets of numbers. A correlation coefficient, written r, is a single number between −1 and +1 that says how closely two variables move together.
| r | Meaning | Example |
|---|---|---|
| +1.0 | Perfect positive: as one goes up, the other goes up in lockstep | Height in inches and height in centimeters |
| About +.70 | Strong positive | Hours studied and practice-test score |
| About 0 | No relationship | Shoe size and anxiety score |
| About −.70 | Strong negative: as one goes up, the other goes down | Anxiety score and hours of sleep |
| −1.0 | Perfect negative | Minutes elapsed and minutes remaining on the exam clock |
Two things to remember. First, the sign tells direction, the size tells strength: a correlation of −.80 is stronger than one of +.40. Second, correlation is not causation. Ice cream sales and drowning deaths rise together every summer, but ice cream does not cause drowning; hot weather drives both. On the exam, any option that concludes one variable "causes" another from a correlation alone is wrong.
Correlation coefficients also appear as reliability and validity numbers in lesson 4: a test-retest reliability of .90 is simply the correlation between the two sets of scores.
How the exam asks about this
Pick the right middle. "A counselor has income data for a community with a few very wealthy families. Which measure of central tendency best describes the typical income?" The median. "Which is most affected by outliers?" The mean. "Which can be used with nominal data?" The mode.
Name the skew. "Most students scored high on an easy quiz, with a few very low scores." Negative skew, and the mean is below the median. Look for where the few extreme scores are; the tail points at them.
Compare spread. "Two classes have the same mean. Class B has a larger standard deviation. What does that tell you?" Class B's scores are more spread out; more students are far from the average.
Interpret a correlation. "A study finds r = −.65 between session attendance and symptom severity." Strong negative relationship: more sessions, lower symptoms. Any option claiming attendance causes lower symptoms goes too far.
Without scrolling up, write everything you remember from this lesson. Then check yourself against the key terms. Recalling the whole lesson once, right after reading it, does more than reading it twice.
Ten more questions from the whole domain, with an explanation after each answer.
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